In this lesson, angles will be examined as drawn in a coordinate plane. This allows for the diagramming of angles which are much larger than or which may be negative. Values of the six fundamental trig functions will be found for such angles. The Lesson:
In a right triangle, one angle is 90º and the side across from this angle is called the hypotenuse. The two sides which form the angle are called the legs of the right triangle. We show a right triangle below. The legs are defined as either “opposite” or “adjacent” (next to) the angle A. Definitions:
We shall call the opposite side “opp,” the adjacent side “adj” and the hypotenuse “hyp.”
In the following definitions, sine is called “sin,” cosine is called “cos” and tangent is called “tan.” The origin of these terms relates to arcs and tangents to a circle. Let's Practice:
For more information on these functions reference the lesson on sine, cosine and tangent.
- sin(A) =
- cos(A) =
- tan(A) =
Three more trigonometric ratios can be defined as the reciprocals of these fundamental ratios. They are cosecant, secant, and cotangent. The ratios are given by the following equations:
- csc(A) =
- sec(A) =
- cot(A) =
- We diagram angles in quadrants two and three. Let A = -150º and B = 150º . Because these angles are in quadrants two and three, we draw a “reference triangle” in each quadrant. Using the fact that these triangles are triangles, we can label the sides. The “reference angle” is the 30º angle in the triangles with vertex at the Origin of the coordinate axes.
For angle A = -150º we use the opp, adj and hyp sides of the reference triangle to calculate the values of the six trig functions:
sin(A) = sin(-150º) = For angle B = +150º , the only difference is that the opposite side is +1 instead of -1. We have:
cos(-150º) = :
csc, sec and cot can be found by inverting these results.
sin(B) = sin(150º) =
Again, csc, sec and cot can be found by inverting these results.
- Diagram 405º showing the reference angle and use the reference triangle to calculate the sin, cos and tan of 405º. We show the diagram below. The opp and adj sides of the reference triangle are both . The hyp is 2. This is true of any triangle.
Using the sides of the reference triangle, we can produce values of trig functions of :
The csc, sec and tan can be found by inverting these values.
- Find the sin and tan of a quadrant three angle with a reference triangle having opposite side -2 and hypotenuse 5. If this angle , what is the measure of this angle?
We have sin(q) = . To find tan(q) we diagram the angle below and use the Pythagorean Theorem to find that the adjacent side has measure - .
We have tan(q) =
We can find the reference angle using on a calculator. We find ref angle . The angle q is . We have .
We can generalize some of these results.
- Notice that the opposite side is negative in quadrants three and four. Therefore the sine which is is negative in quadrants three and four but positive in quadrants one and two.
- Notice that the adjacent side is negative in quadrants two and three. Therefore the cosine which is is negative in quadrants two and three and positive in quadrants one and four.
- The tangent will be positive when the opposite and adjacent sides have the same sign because the tangent is . This occurs in quadrants one and three.
One way to remember which trig functions are positive in each quadrant is illustrated in the diagram below.
Using the letter in the diagram as a device for remembering positive values, we have “All Students Take Calculus” reminding us of the basic trig functions which are positive in each quadrant: All are positive in quadrant one; Sine is positive in quadrant two; Tangent is positive in quadrant three; and Cosine is positive in quadrant four. They are negative elsewhere, which would be obvious if a good diagram is labeled showing the possible negative values of the opposite or adjacent sides. (Note that the hypotenuse is the absolute value of the distance from the origin and is always positive).